Orbital Mechanics and Kepler's Laws

An orbit is a continuous free-fall. The satellite falls toward Earth, but moves forward fast enough that the curved Earth "falls away" at the same rate. The satellite is always falling but never hits the surface.

This is why astronauts feel weightless — they and the ISS are falling together.

These were discovered by Kepler (1609–1619) from observational data, before Newton explained the cause.

Law 1 — Elliptical Orbits

Planets orbit the Sun in ellipses, with the Sun at one focus.

A circle is a special case (both foci coincide). Most planetary orbits are nearly circular with small eccentricity.

Law 2 — Equal Areas

A line from a planet to the Sun sweeps equal areas in equal time intervals.

Consequence: planets move faster when closer to the Sun (perihelion) and slower when farther (aphelion). Angular momentum is conserved — L = mvr = constant.

Law 3 — Period–Radius Relation

T² ∝ a³ (period squared is proportional to the cube of the semi-major axis)

For circular orbits: T² = (4π²/GM) × r³

This means if you know any planet's orbital radius, you can find its year, and vice versa.

Circular Orbital Speed

v = √(GM/r)

Speed decreases as radius increases. Outer planets orbit more slowly.

Time Period

T = 2πr/v = 2π√(r³/GM)

Escape Velocity

v_esc = √(2GM/r) = √2 × v_orbital

Escape velocity is always √2 ≈ 1.41 times the orbital velocity at the same radius.

At Earth's surface: v_esc ≈ 11.2 km/s, v_orbital (low Earth) ≈ 7.9 km/s.

Total Energy of a Satellite (Elliptical Orbit)

E = -GMm / (2a)

Negative energy means the satellite is bound. As a increases (larger orbit), total energy becomes less negative (closer to zero = escape).

Vis-Viva Equation (works at any point in ellipse)

v² = GM(2/r − 1/a)

Where r is current distance, a is semi-major axis. At perihelion (r = r_min), speed is maximum.

Geostationary: same angular velocity as Earth's rotation → appears stationary from ground. Must be above the equator.

Gravity = Centripetal force:

GMm/r² = mv²/r → v² = GM/r

Period T = 2πr/v → v = 2πr/T

Substituting:

(2πr/T)² = GM/r T² = 4π²r³/GM

So T² ∝ r³ (Kepler 3 derived). The constant 4π²/GM depends only on the central body (Sun for planets).

• Weightlessness in orbit = free fall, not absence of gravity
• v ∝ 1/√r — memorise: doubling radius → speed drops by factor √2
• T ∝ r^(3/2) — doubling radius → period increases by factor 2√2 ≈ 2.83
• Escape velocity derivation: set KE = gravitational PE → v = √(2GM/r)
• Angular momentum conserved in orbit: r₁v₁ = r₂v₂ (useful for Kepler 2)
• Total energy = KE + PE = GMm/2r − GMm/r = −GMm/2r (for circular)
• Binding energy = |E| = GMm/2r — energy needed to escape the orbit